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Peanoâ•Žs Arithmetic Proceedings of GREAT Day Volume 2014 Article 10 2015 Peano’s Arithmetic Carmen Staub SUNY Geneseo Follow this and additional works at: https://knightscholar.geneseo.edu/proceedings-of-great-day Creative Commons Attribution 4.0 License This work is licensed under a Creative Commons Attribution 4.0 License. Recommended Citation Staub, Carmen (2015) "Peano’s Arithmetic," Proceedings of GREAT Day: Vol. 2014 , Article 10. Available at: https://knightscholar.geneseo.edu/proceedings-of-great-day/vol2014/iss1/10 This Article is brought to you for free and open access by the GREAT Day at KnightScholar. It has been accepted for inclusion in Proceedings of GREAT Day by an authorized editor of KnightScholar. For more information, please contact [email protected]. Staub: Peano’s <i>Arithmetic</i> Peano’s Arithmetic Carmen Staub y the middle of the nineteenth century, many back to the publisher with corrections and his own concerns regarding the foundation of math- suggestions for improvement. He even asked for per- ematics began to arise in Europe. Mathema- mission to publish Genocchi’s lectures. After fixing Bticians questioned the process of deriving theorems; some errors and adding his own comments to the how could one eliminate human intuition from collection, he only listed himself as an editor [2]. proofs? There had not been a set process that would guarantee their validity. There needed to be a type of In 1884 Peano became a professor at the university. systemization that would take care of defining no- In the five years that followed, Peano produced many tation and the axioms that would serve as the basis significant mathematical results. For example, he for all proofs. The first area to start in was arithme- proved that if a function f(x, y) is continuous, then tic. In 1889, Giuseppe Peano published Arithmet- the first order differential equationdx/dy = f(x, y) has ices principia, nova methodo exposita, in an attempt a solution [4]. But Peano’s most well-known contri- to construct a well-defined system of arithmetic with bution to mathematics was his axioms on the natural concrete axioms. Peano’s work was thorough yet sim- numbers featured in his Arithmetices principia. ple; thus it was adopted by other mathematicians and However, Peano was not the first person to attempt grew to become the official system of arithmetic used this. Other mathematicians, such as Hermann Grass- today. mann and Gottlob Frege, had already been working Giuseppe Peano was born on August 27, 1858 near to systemize arithmetic and its axioms. Grassmann, Cuneo, in the region of Piedmont, Italy. He was one in fact, was the first to begin this process in his book of five children whose parents were farmers. He went Lehrbuch der arithmetik (1861). Frege made more to school in Spinetta until he was twelve years old progress on this and focused more on the logic be- when his uncle, who recognized his talent, brought hind it in his Begrifftschrift(1879) [1]. Peano studied him to Turin. Peano received private lessons from his these texts, along with some others, before he began uncle until he was qualified to enroll in the Cavour his Arithmetices principia. His goal was to set up a sol- School, a secondary school in the city. He graduated id system of arithmetic, improve logic symbols and from the high school in 1876 with a college scholar- notation, and establish axioms that would serve as ship and, in the same year, he entered the University the basis for all arithmetic results. Peano believed that of Turin. ordinary language—and therefore any mathematics that was explained in it—was too ambiguous. He While the majority of his peers were studying engi- fought to fine tune the details. To do this, he wrote neering, Peano was one of the only students to study his axioms, definitions, and proofs entirely in the pure mathematics. He worked closely with several symbols that he defined in the preface of Arithmetices professors, including Enrico D’Ovidio and Angelo principia. The idea was that with this structure, every Genocchi. After graduating in 1880 with high hon- result in arithmetic could be derived [3]. The thirty- ors, Peano went back to be an assistant for both pro- six page booklet was published in 1889. fessors [4]. Peano begins the book by listing signs of logic and Peano was well known for his extreme rigor in his arithmetic that he claims to be sufficient enough to work in mathematics. While he was working as a express any mathematical proposition. While most teacher’s assistant, he would often send textbooks of the symbols have been previously developed and 96 Published by KnightScholar, 2015 1 Proceedings of GREAT Day, Vol. 2014 [2015], Art. 10 used by other mathematicians, Peano did introduce Lastly, Peano includes in his list of axioms a setup for several symbols. He used to signify “and” between induction: two propositions, and to mean “or”. The rotated ɛ ɛ x ɛ k : Ɔ . x + 1 ɛ k Ɔ. C, or Ɔ was used to mean∩ either “one deduces” or “is 9. k K 1 K x contained in,” depending∪ on the context in which it N Ɔ k. [2] ∴ ∴ x ɛ N . ∷ is used. While the former definition is used in im- Here k is a class (or we can think of it as a property) plications, the latter is used with sets being in other and 1 is k. Also x is a positive integer. If x is k then x + sets. Another symbol he is known for is ɛ, read as “is,” 1 is k, then all natural numbers are contained in class and is an early version of , which now has a more k. This is the outline of the induction that is widely specific definition to signify membership of a set [6]. used today. Let k be what we want to prove for all ∈ While many of us include zero in the natural num- natural numbers. The base case is that 1 works for k. bers when working with arithmetic, Peano does not Next we show that if x works for k then x + 1 works mention it in his original postulates. Instead of the for k, then for all natural numbers, k holds [2]. symbol , which we now use to represent the natu- Immediately after these nine axioms, Peano begins a ral numbers, Peano uses the symbol N for positive long list of definitions and proofs that he has com- ℕ integers, or simply “numbers” as he calls them. Fol- pletely written in his aforementioned symbols. For lowing the preface is an introduction on logical nota- example, he starts out defining the other numbers in tion. Peano goes over punctuation, propositions, and this style: propositions of logic, classes, the inverse, and func- tions [2]. DEFINI TIONS After that, Peano lists his famous axioms in this way: 10. 2 = 1 + 1; 3 = 2 + 1; 4 = 3 + 1; etc. 1. 1 ɛ N. This is read as “1 is a positive integer,” or rather “1 is THEOREMS a natural number.” Numbers 2, 3, 4, and 5 show that 11. 2 ɛ N the equality relation is reflexive, symmetric, transi- tive, and closed, respectfully: PROOF 2. a ɛ N. Ɔ . a = a. 1 ɛ N Axiom 1 3. a, b ɛ N. Ɔ : a = b . = . b = a. 1 ɛ N . Ɔ . 1 + 1 ɛ N Axiom 6 4. a, b, c ɛ N. Ɔ a = b . b = c. : Ɔ . a = c. 1 + 1 ɛ N 5. a = b . b ɛ N : Ɔ . a ɛ N. ∴ 2 = 1 + 1 Definition 10 Numbers 6, 7 and 8 deal with the successors: 2 ɛ N (Theorem) [2] 6. a ɛ N. Ɔ . a + 1 ɛ N. Of course, this is the simplest of the proofs in the For each natural number there is a successor. booklet. However, the others do not get much more complex. The following is taken from the section on 7. a, b ɛ N. Ɔ : a = b . = . a + 1 = b + 1 . multiplication: If two natural numbers are the same, then their suc- cessors are the same. DEFINI TIONS 8. a ɛ N. Ɔ . a + 1– = 1. 1. a ɛ N . Ɔ . a × 1 = a. If a is a natural number, then a + 1 cannot equal 1. 2. a, b ɛ N . Ɔ . a × (b + 1) = a × b + a . ab = a × b; ab In other words, 1 is not the successor of any number. + c = (ab) + c; abc = (ab)c. 97 https://knightscholar.geneseo.edu/proceedings-of-great-day/vol2014/iss1/10 2 Staub: Peano’s <i>Arithmetic</i> THEOREMS explicitly define the successor function in an axiom. 3. a, b ɛ N. Ɔ . ab ɛ N. Also, Dedekind denotes the successor of a number a as a' while Peano uses a +1 which makes the defini- tion more obvious [7]. PROOF After the publication of Arithmetices principia, math- a ɛ N . P1 : Ɔ : a × 1 ɛ N : Ɔ . 1 ɛ [bɛ]Ts. (1) ematicians adopted Peano’s axioms and made a few a, b ɛ N . b ɛ [bɛ]Ts : Ɔ : a × b ɛ N . §1P19 : Ɔ : ab modifications, such as including zero to form the + a ɛ N. modern natural numbers [5]. Bertrand Russell took his axioms and tried to apply them to more general, P1 : Ɔ : a(b + 1) ɛ N : Ɔ : b + 1 ɛ [bɛ]Ts. (2) philosophical concepts beyond the natural numbers. In fact, In fact, in more than one interview, he states (1)(2) . Ɔ . Theor. [2] that Peano was his inspiration to start closely examin- While this example may seem more intimidating at ing mathematical logic: first, once one deciphers the symbols it is merely a It was at the International Congress of quick proof by induction.
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